Homework Assignment 6 in Differential Equations, MATH308
due March 21, 2012
Topics covered : definition and properties of Laplace transform; inverse Laplace transform of rational
functions; solution of initial value problems using Laplace transform; step function and Laplace transform
of discontinuous functions (corresponds to sections 6.1, 6.2, 6.3 in the textbook)
1. Recall that the hyperbolic cosine cosh t and hyperbolic sine sinh t are defined as follows:
cosh t =
et + e−t
,
2
sinh t =
et − e−t
.
2
Using the definition of the Laplace transform, find the Laplace transform of the given function
(below a and b are real constants):
(a) f (t) = sinh bt;
(b) f (t) = eat cosh bt
(show your work).
2. Find the inverse Laplace transform of the given function:
3s
;
s2 + 2s − 8
2s + 5
(b) F (s) = 2
s + 6s + 25
(a) F (s) =
3. Solve for Y (s), the Laplace transform of the solution y(t) to the given initial value problem (you
do not need to find the solution y(t) itself here):
(a) y 00 − 3y 0 + 2y = cos t,
(b) y 00 + y 0 − y = t3 ,
y(0) = 0, y 0 (0) = −1;
y(0) = 1, y 0 (0) = 0
4. Using the method of Laplace transform solve the following initial value problem:
y 00 + 6y 0 + 5y = 12et ,
y(0) = −1, y 0 (0) = 7.
5. (a) Find the Laplace transform of the function


0
f (t) = t


1
t < 1,
1 ≤ t < 2,
2 ≤ t.
(b) Find the inverse Laplace transform of the function
1
e−2s − 3e−4s
.
s+2